Problem 46938. Numerical computation of the optimal shooting angle of a catapult

Created by Brandon Caasenbrood

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Consider a capapult that fires a projects into the air with an initial velocity. The free-flying projectile is subjected to air friction and a gravitional force. Given a desired target  and an initial velocity , find the optimal shooting angle of the catapult that minimizes the distance between the target and the trajectory of the fired projectile. 
tip 1: Consider the states  and  as the x- and y-position of the projectile, and  and  as the x- and y-velocity. Then, the trajectory of the projectile can be found by solving the following ordinary differential equation (ODE):
        ,     
              
        ,      
        .   
where , and is the friction coefficient between the air and the projectile. Use the ode45.m function to compute the trajectory of the projectile with initial conditions . Plotting  vs.  will result in the x-y trajectory of the projectile, as shown in the figure below.
tip 2: Use the following update law, to incrementally update the shooting angle :
         
where  the smallest Euclidean distance between the trajectory of the projectile and the target , is a difference angle, and an update parameter.
Example of algorithm's numerical result:
theta = catapult(25,3,25)
theta = 
    0.8431
    

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35 Solutions
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Last Solution submitted on Jan 02, 2025

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2 Comments

Alex on 19 Oct 2020

Nice problem. There should be much more problems involving solution of differential equations on Cody. :-)

William on 7 Mar 2024

I have tried many ways of getting isequal( round(...,2),y_correct) to evaluate to 'true' when round(...,2) is, in fact, equal to y_correct. But, they all fail on the Cody server. Why not just test whether abs(catapult(...) - y_correct) <= TOL.

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